Systematic derivation of a rotationally covariant extension of the two-dimensional Newell-Whitehead-Segel equation.

An extension of the Newell-Whitehead-Segel amplitude equation covariant under abritrary rotations is derived systematically by the renormalization group method. Typeset using REVTEX 1 The present day theory of pattern formation to a large degree rests on the derivation and exploitation of amplitude equations. In particular, for the formation of 2-dimensional patterns by spontaneous symmetry breaking at a finite wavenumber in isotropic 2-dimensional layers the amplitude equation due to Newell and Whitehead and to Segel [1] is of fundamental importance. It is derived, e.g. for the Benard instability, close to the onset of the pattern, by assuming the presence of an ideal one-dimensional roll pattern and certain scaling properties of its small amplitude and its spatial and temporal variations. The small parameter is the difference of the bifurcation parameter from its value at onset. The resulting amplitude equation, despite of its great fundamental and practical importance, has a well known short-coming: it does not respect the full rotation invariance of the two-dimensional system and the fact, that in principle the rotational symmetry is broken by the pattern spontaneously rather than by any external agent. This short-coming is a direct consequence of the method of derivation, which singles out a particular direction for the main pattern, and even more importantly, makes use of anisotropic assumptions for the scaling of the spatial variation of the pattern. Two recent developments now permit to overcome this problem [2]. First, Gunaratne et al [3] recently proposed an extended form of the Newell-Whitehead-Segel (N-W-S) equation which they demonstrated to fully respect the rotational symmetry of the original system. They also provided a derivation of their equation by including, in a given order in the bifurcation parameter, symmetry restoring terms, which, indeed appear in higher order of the N-W-S scheme. While this recombining of terms from different orders of the expansion suggests that their equation is the correct symmetric extension of the N-W-S equation, before one can accept this as a fact, a fully systematic (if not mathematically rigorous) derivation